, Volume 80, Issue 4, pp 1214–1277 | Cite as

Simultaneous Embedding: Edge Orderings, Relative Positions, Cutvertices

  • Thomas BläsiusEmail author
  • Annette Karrer
  • Ignaz Rutter


A simultaneous embedding (with fixed edges) of two graphs \(G^{\textcircled {1}}\) and \(G^{\textcircled {2}}\) with common graph \(G=G^{\textcircled {1}} \cap G^{\textcircled {2}}\) is a pair of planar drawings of \(G^{\textcircled {1}}\) and \(G^{\textcircled {2}}\) that coincide on G. It is an open question whether there is a polynomial-time algorithm that decides whether two graphs admit a simultaneous embedding (problem Sefe). In this paper, we present two results. First, a set of three linear-time preprocessing algorithms that remove certain substructures from a given Sefe instance, producing a set of equivalent Sefe instances without such substructures. The structures we can remove are (1) cutvertices of the union graph \(G^\cup = G^{\textcircled {1}} \cup G^{\textcircled {2}}\), (2) most separating pairs of \(G^\cup \), and (3) connected components of G that are biconnected but not a cycle. Second, we give an \(O(n^3)\)-time algorithm solving Sefe for instances with the following restriction. Let u be a pole of a P-node \(\mu \) in the SPQR-tree of a block of \(G^{\textcircled {1}}\) or \(G^{\textcircled {2}}\). Then at most three virtual edges of \(\mu \) may contain common edges incident to u. All algorithms extend to the sunflower case, i.e., to the case of more than two graphs pairwise intersecting in the same common graph.


Simultaneous planarity Efficient algorithm Graph drawing 



We thank the anonymous reviewers for thoroughly reading an earlier version of this paper and for providing useful comments, which helped us to to improve the presentation of our paper.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Hasso Plattner InstitutePotsdamGermany
  2. 2.Karlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.TU EindhovenEindhovenThe Netherlands

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